J. Sethuraman and R. C. Tiwari. Convergence of Dirichlet measures and the interpretation of their parameter. In S. S. Gupta and J. O. Berger (Eds.), Statistical Decision Theory and Related Topics III, pages 305-15, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....construction p 1 = V 1 and p k = 1 V 1 ) 1 V 2 ) 1 V k 1 ) V k k = 2; N (3) where V 1 ; V 2 ; VN 1 are i.i.d. Beta(1; random variables and we set VN = 1 to ensure that P N k=1 p k = 1. By the construction given in Sethuraman (1994) see also McCloskey 1965; Sethuraman and Tiwari 1982; Donnelly and Joyce 1989; Perman, Pitman and Yor 1992) it easily follows that PN converges almost surely to a Dirichlet process with measure H , written as DP( H) i.e. PN a:s DP( H) We refer to H as the reference distribution and as the Dirichlet mass parameter. See also Muliere and ....

Sethuraman, J. and Tiwari, R. C. (1982). Convergence of Dirichlet measures and the interpretation of their parameters. Statistical Decision Theory and Related Topics III 2 305-315.

....distribution. The scale parameter c is used as a tuning constant: it acts like a prior sample size using the heuristic of a prior distribution being equivalent to a data set that is in effect merged with the y i and c can be chosen for good coverage behavior of interval estimates. Sethuraman and Tiwari (1982) detail how to simulate from a Dirichlet process. If F D(cF 0 ) we can generate an F via F = 1 X j=1 V j ffi j ; 16) where V 1 = W 1 ; V j = W j (1 Gamma W j Gamma1 ) 1 Gamma W 1 ) j = 2; 3; W 1 ; W 2 : are IID Beta(1; c) and 1 ; 2 ; are IID from F 0 . In ....

Sethuraman J, Tiwari R (1982). Convergence of Dirichlet measures and the interpretation of their parameters. In Proceedings of the Third Purdue Symposium on Statistical Decision Theory and Related Topics, SS Gupta, JO Berger (eds.), New York: Academic Press.

....according to the Dirichlet process make it unsuitable as a way to model smooth densities. However, it can be connected to the infinite normal mixture model previously set up, as noted by Ferguson (1983) and Lo (1984) We will connect the density model to the Dirichlet process using a result due to Sethuraman and Tiwari (1982) and Sethuraman (1994) See also Florens and Rolin (1994) Let 1 ; 2 ; be independent draws from the normalized Dirichlet base measure = Theta) Let r 1 ; r 2 ; be independent draws from Beta(1; Theta) a beta distribution with parameters 1 and ( Theta) Form p j = r j Q j ....

Sethuraman, J., and R. C. Tiwari (1982): "Convergence of Dirichlet Measures and the Interpretation of Their Parameter," in Statistical Decision Theory and Related Topics III, in two volumes, vol. 2, pp. 305-- 315.

....See Ferguson and Klass (1972) An unpublished thesis by McCloskey (1965) appears to be the first work that drew comparisons between the Dirichlet process and beta random variable stick breaking procedures. However, it wasn t until Sethuraman (1994) that these connections were formalized. Also see Sethuraman and Tiwari (1982), Donnelly and Joyce (1989) and Perman, Pitman and Yor (1992) Let Gamma k = E 1 Delta Delta Delta E k , where E k are iid exp(1) random variables. Sethuraman (1994) established the following remarkable identity showing that the Dirichlet process defined by Ferguson (1973) is a Ishwaran ....

Sethuraman, J. and Tiwari, R. C. (1982). Convergence of Dirichlet measures and the interpretation of their parameters. Statistical Decision Theory and Related Topics III 2 305-315.

....1 n n X i=1 ffi X i : 8 The weights given to the prior mean and the empirical distribution are the ratio of the total mass of ff, ff(X ) and the sample size, n. Hence, ff(X ) can be viewed as the prior sample size. For this reason, ff(X ) 0 is sometimes considered to be noninformative, but Sethuraman and Tiwari (1982) indicated possible dangers in this notion of a noninformative Dirichlet process. Theorem 4 Let ff 0 ; ff 1 ; Delta Delta Delta be a sequence of finite measures on X such that lim r 1 ff r (X ) 0 lim r 1 sup A2B j ff r (A) ff r (X ) Gamma ff 0 (A) ff 0 (X ) j = 0: Then D ff r ....

Sethuraman, J. and Tiwari, R. C. (1982), "Convergence of Dirichlet Measures and the Interpretation of Their Parameter," in Statistical Decision Theory and Related Topics III, in two volumes, vol. 2, pp. 305-- 315.

....a particular conditional distribution given P . But a conditional distribution of P given r( P ) that serves for some sequence P with a prescribed distribution must work for every sequence P with that distribution. 2 Combining Theorems 3 and 5 yields the following result: Corollary 8 [64, 63]. Let F be defined by F = 1 X j=1 P j ffi ( X j ) 9) for two sequences of random variables ( P j ) and X j such that ( P j ) has the gem ( distribution (8) and the X j are i.i.d ( independent of ( P j ) Then F has dirichlet( distribution. This construction ....

J. Sethuraman and R. C. Tiwari. Convergence of Dirichlet measures and the interpretation of their parameter. Statistical Decision Theory and Related Topics III, 2:305--315, 1982.

....surely discrete. Thus, using ffi x to denote a point mass at x, we can write G( P 1 j=1 w j ffi j j ( A constructive definition of G DP (ffG) describes how the weights w j and the point masses are sampled: j j G, independently, and w j Beta(ff; 1) Theta Q i j (1 Gamma w i ) see Sethuraman and Tiwari (1982). ii) Let k denote the number of distinct i among f 1 ; n g in (6) Given k, model (6) reduces to a mixture of k normals, i.e. k can be thought of as the size of the mixture (6) Asymptotically, as n 1, k has an a priori mean of ff log(n) iii) The random measure G generated by ....

SETHURAMAN, J. & TIWARI, R.C. (1982). Convergence of Dirichlet measures and the interpretation of their parameter In Statistical Decision Theory and Related Topics III, vol 2, Ed. S.S. Gupta and J.O. Berger, pp. 305--15. New York: Springer-Verlag.

.... this process is a positive finite measure m on the real line, formed from a positive constant a and a distribution function G by m( Gamma1; t] aG(t) Rather than defining the Dirichlet Binary regression 7 process by finite dimensional distributions, we present a constructive definition (Sethuraman and Tiwari 1982). Such a random element of F is, for all t 2 R, F (t) 1 X i=1 w i 1[v i t] where v i are iid G, and the weights w i are the result of a stick breaking exercise based on a set of iid Beta random variables b i having density proportional to (1 Gamma b) a Gamma1 . Precisely, w 1 = b 1 , ....

Sethuraman, J. and R. C. Tiwari (1982), "Convergence of Dirichlet measures and the interpretation of their parameter", Stat'l. Decision Th. and Related Topics III, in two volumes; Shanti S. Gupta, James O. Berger (Ed); Academic Press, 3(2) 305--315.

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J. Sethuraman and R. C. Tiwari. Convergence of Dirichlet measures and the interpretation of their parameter. In S. S. Gupta and J. O. Berger (Eds.), Statistical Decision Theory and Related Topics III, pages 305-15, 1982.

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J. Sethuraman and R. C. Tiwari. Convergence of Dirichlet measures and the interpretation of their parameter. In S. S. Gupta and J. O. Berger (Eds.), Statistical Decision Theory and Related Topics III, pages 305-15, 1982.

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Sethuraman, J. and Tiwari, R.C. (1982). Convergence of Dirichlet measures and the interpretation of their parameter. In Statistical Decision Theory and Related Topics III 2, eds. Gupta and Berger, New York: Academic Press.

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Sethuraman, J. and Tiwari, R.C. (1982). Convergence of Dirichlet measures and the interpretation of their parameter. In: Statistical Decision Theory and Related Topics III, Eds: Gupta, S. and Berger, J.O., Springer-Verlag, New York, 2, 305315.

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Sethuraman, J. and Tiwari, R.C. (1982). Convergence of Dirichlet measures and the interpretation of their parameter. Statistical Decision Theory and Related Topics III, Vol. 2, eds. S.S. Gupta and J.O. Berger, Academic Press, 305-315.

No context found.

Sethuraman, J. and Tiwari, R.C. (1982). Convergence of Dirichlet measures and the interpretation of their parameter. In Statistical Decision Theory and Related Topics III 2, eds. Gupta and Berger, New York: Academic Press.

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Sethuraman, J. and Tiwari, R. (1982). Convergence of Dirichlet measures and interpretation of their parameters. In Statistical Decision Theory and Related Topics. III 2 (Gupta, S. S. and Berger, J. O., Eds.), Academic Press, New York, 305--315.

No context found.

Sethuraman, J., and Tiwari, R. C. (1982), "Convergence of Dirichlet Measures and the Interpretation of Their Parameter," Stat'l. Decision Th. and Related Topics III, in two volumes; Shanti S. Gupta, James O. Berger (Ed); Academic Press, 3(2), 305--315.

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